92 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

( xvF ), is in disagreement with experiment [161]. Thus, in short, the RVB picture allows us to connect d–wave superconductivity and some basic features of the cuprate phase diagram to the insulating Mott state in a smooth way and the corresponding mean field theory yields reasonable results. On the other hand, the RVB theory is di cult to handle owing to constraints, and the mean–field theory is di cult to control. Finally, we stress that in contrast to our perturbative paramagnon–like theory, there are only a few reliable results for excitations and dynamical properties using the RVB picture [162], because projected wave functions can only treat statics so far. We believe, in view of the detailed comparison with experiments in the next chapter for the elementary excitations (ARPES) and their interdependence with spin excitations (INS experiments), that this is probably the main advantage of our theory.

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100 3 Results for High–Tc Cuprates: Doping Dependence

of a Cooper pair. Thus, in conventional superconductors, Tc is proportional to ∆(T = 0) [5]. In contrast to this, the observation Tc ns in underdoped (hole–doped) cuprates indicates that Cooper pair phase fluctuations drive the superconducting instability. The Cooper pairs break up only at a crossover temperature Tc > Tc, and between Tc and Tc local Cooper pairs exist, but without long–range phase coherence [2, 3, 4, 6, 7].

To remind the reader, there exists a third, even higher temperature scale T , below which a pseudogap starts to form, as seen in NMR, tunneling spectroscopy, electronic transport, and Hall e ect measurements (to name

just a few) [8, 9, 10, 11, 12, 13, 14]. The regions Tc < T < Tc and Tc < T < T are often called the strong and weak pseudogap regimes, respectively.

We shall demonstrate that our electronic theory that assumes the exchange of antiferromagnetic spin fluctuations as the relevant pairing mechanism for singlet pairing in cuprates can account for the main features in the phase diagram of hole–doped cuprates. In particular, we determine the doping dependence of the relevant temperatures of the phase diagram, namely Tc (x), Tc(x), and also T , at which a gap appears in the spectral density. Below Tc we indeed find incoherent Cooper pairs (“preformed pairs”), which become phase–coherent only below the critical temperature Tc of the bulk material. We show that phase fluctuations, contributing ∆Fphase to the free energy, lead to a decreasing critical temperature in the underdoped regime and thus to the appearance of an optimal doping xopt. It is shown that this result is due to the small superfluid density ns(T ) in the system. Most importantly, we calculate that ∆Fcond > ∆Fphase (where ∆Fcond denotes the contribution to the free energy due to Cooper pair formation, where ∆Fphase denotes the contribution due to phase fluctuations of the Cooper pairs), for a doping x < xopt, and ∆Fcond < ∆Fphase for x > xopt. We compare our results with the Berezinskii–Kosterlitz–Thouless (BKT) theory and with the XY model and find similar results for the resulting phase diagram. Finally, we also discuss the relaxation dynamics in pump–probe spectroscopy and find reasonable agreement with experiment. Of course, a detailed quantitative comparison for all classes of cuprate superconductors within the simple two–dimensional one–band Hubbard model is beyond the scope of this book. Nevertheless, we shall show that the key facts can be explained within our approach.

In Fig. 3.1, results are shown for ∆F (x). We find that ∆Fcond mainly follows the doping dependence of the mean–field transition temperature Tc . On the other hand, as discussed in the previous chapter, the doping dependence of ns(0)/m determines the doping dependence of ∆Fphase . Thus the energy cost due to phase fluctuations has the opposite behavior to the energy gain due to Cooper pair condensation with respect to the doping concentration x. It is remarkable that we obtain from our electronic theory a crossing of the two energy contributions ∆Fcond and ∆Fphase at x 0.15, where the largest Tc is observed. The consequence of this is that we find theoretically Tc ns

Fig. 3.1. Calculated crossover of the phase–sti ness energy. We find ∆Fphase ns /m , whereas the condensation energy ∆Fcond α{ns /m}∆0(x). Here, we estimate α 1/400. Note that ∆Fphase < ∆Fcond implies two characteristic temperatures: Tc , where Cooper pairs are formed at Tc ∆0, and Tc ≈ ∆Fphase ns, where Cooper pairs become phase–coherent.

for underdoped cuprates (and thus the Uemura scaling), and a nonmonotonic doping dependence of Tc(x) with an optimal doping at x 0.15. Physically speaking, in the overdoped regime we find a large ∆Fphase which means that Cooper pair phase fluctuations are associated with a large amount of energy. Thus the system will undergo a mean–field transition because of the small condensation energy ∆Fcond. In the underdoped regime of cuprate superconductors, the situation is the opposite: the energy gain due to the formation of Cooper pairs is not large enough to reach the Meissner state of the bulk material. This is only possible at a smaller temperature, where the Cooper pairs become phase–coherent; this temperature is determined by ∆Fphase and

∆Fphase < ∆Fcond.

Thus we can safely conclude that in the overdoped regime, Tc is identical to the bulk transition temperature Tc below which a Meissner e ect is found experimentally. Further evidence for a mean-field superconducting transition comes from the fact that Tc ∆(T → 0), which we have calculated within our electronic theory. In contrast to this, in the underdoped regime the system behaves more two-dimensionally and thus, owing to the short coherence length of a Cooper pair in cuprates, another energy scale, namely the small superfluid density ns, becomes important and leads to the fact that Tc < Tc . The temperature range Tc < T < Tc may be viewed as the region where local Cooper pairs without long–range phase coherence (“preformed pairs”) can exist. The occurrence of preformed pairs was postulated by Chakraverty and coworkers [3] and later by Emery and Kivelson [4]. Our calculations provide a microscopic justification for this scenario. However, no clear experimental proof of the existence of preformed pairs has been obtained so far.

102 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.2. Phase diagram for high–Tc superconductors resulting from considering a spin–fluctuation–induced Cooper pairing, including phase fluctuations. The calculated values for ns(0)/m are in good agreement with muon spin rotation experiments [15]. Tc denotes the temperature below which Cooper pairs are formed. The dashed curve gives the observed Uemura scaling Tc ns(T = 0, x) [1]. Below T we obtain a gap structure in the spectral density, as observed in tunneling spectroscopy [16, 17]. The solid curve Tcexp, which describes many hole–doped superconductors, is taken from [18, 19].

In order to summarize our calculations, we now show our results for the resulting phase diagram for hole–doped cuprates in Fig. 3.2. Note that for illustration we have added the experimental Tc(x) curve, which describes many hole-doped superconductors as pointed out by Tallon and coworkers [18, 19]. As mentioned earlier, the superconducting (mean–field) transition temperature Tc , below which one finds a finite gap function, has been determined from the linearized version of the gap equation (see (A.18)).

In order to illustrate the important behavior of ns in more detail, we show in Fig. 3.3 the temperature dependence of ns(ω = 0)/n (n denotes the normal–state band filling) below Tc for various doping concentrations. For this purpose we have calculated the current–current correlation function using standard many–body theory [20] and taken the corresponding Green’s functions within the FLEX approximation. This has been described in the previous chapter. Note that according to London’s theory [21], which states that λL ns, the ratio ns/n can be related to measurements of the (in–plane) penetration depth, for example in microwave experiments.

As can also be seen from Fig. 3.3, qualitative agreement with the data of Bonn, Hardy and coworkers on λ2(T = 0)/λ2(T ) concerning the slope of the curves in the vicinity of Tc and the linear behavior for T → 0 is found [22]. In particular, the FLEX approximation to the generalized Eliashberg equations yields, close to Tc , a relation λ3(T = 0)/λ3(T ) (Tc − T ). The same power law has been found by Kamal et al. and has been attributed to critical fluctuations starting about 10 K below Tc, since the slope coincides with the

104 3 Results for High–Tc Cuprates: Doping Dependence

critical exponent for the 3D XY model [22]. Here, we obtain this power law from the generalized Eliashberg equations using the FLEX approximation, which is purely 2D and does not contain critical fluctuations. Instead, the rapid increase of ns below Tc is due to the self-consistent treatment of the superconducting gap function ∆(ω). Thus, we conclude that while 3D critical fluctuations are expected in a very narrow temperature range close to Tc, they are not the origin of the observed power law on the scale of 10 K.

Note that our calculations also show that roughly one–third of the holes become superconducting, even for T → 0. This is typical of a strongly interacting system and is further support for the suggestion that preformed pairs might exist in the underdoped regime.

Comparison with BKT theory

Shortly after the discovery of cuprate high–Tc superconductors, many experiments were interpreted in term of the BKT theory for bulk samples [23, 24, 25, 26, 27, 28]. Recently, an important experiment has been performed by Xu et al. who have found signs of vortices at temperatures much higher than Tc in underdoped La1−xSrxCuO4 in measurements of the Nernst e ect [29]. The most recent reanalysis of their data gives an onset temperature for vortex e ects of 40 K for an extremely underdoped sample with a doping x = 0.05, and even 90 K for x = 0.07 [30].

As an example, we show in Fig. 3.3 the superconducting bulk transition temperature Tc for an underdoped cuprate (dashed line). ns(x, T )/m has been taken from the solutions of the generalized Eliashberg equations. Thus, in the underdoped regime one indeed finds a di erence between Tc and Tc . As already mentioned above, a finite value of ns(Tc < T < Tc ) can be interpreted in terms of local Cooper pairs with a strongly fluctuating phase. In the case of YBa2Cu3O6+x (YBCO), this has been recently confirmed by experiment [33].

In Fig. 3.4a results are given for ns(T, x)/m, where m is the e ective mass. We again find that Cooper pair phase fluctuations are unimportant in the overdoped regime. Note that ns(T, x) → 0 for T → Tc , since Cooper pairs disappear at Tc . However, the phase coherence temperature Tc has to be determined by spatially averaging over the Cooper pair phase fluctuations. In a Ginzburg–Landau (GL) treatment, the phase information is given by the

Tc so that no Meissner e ect occurs above Tc. The transition temperature Tc has to be determined by taking Cooper pair phase fluctuations into account. The BKT theory, as well as the 2D [32] and 3D XY models [31], predicts that Tc is proportional to ns(Tc)/m. As discussed above, in BKT theory the phase sti ness, which is proportional to ns/m as shown in (2.134), assumes the value a = 2/π at T = Tc. Above Tc the renormalized phase sti ness drops to zero owing to the appearance of free vortices. The free vortices also destroy

x = 0.221 (open circles), x = 0.201 (filled circles), x = 0.188 (triangles), x = 0.155 (squares), x = 0.122 (diamonds), x = 0.091 (open circles). ns vanishes at T = Tc . Tc is obtained from the intersection of ns (x, T ) with aT as indicated, where 1/a = 2.202 (3D XY) [31], 1/a = π/2 (BKT theory), and 1/a = 0.9 (2D XY) [32]. The results obtained for Tc are shown in (b). For comparison, the mean–field result Tc is also displayed.

the Meissner e ect, since the energy needed to create an additional vortex at the edge of the system vanishes. Nevertheless, the local superfluid density ns remains nonzero up to Tc . For the 2D XY model, the corresponding value is a ≈ 1/0.9 [32] and in the 3D XY model it is a ≈ 1/2.202 [31]. These three Tc criteria correspond to the intersections of the three straight lines in Fig. 3.4a with the curves of ns(T )/m. The resulting values for Tc(x) are shown in Fig. 3.4b. Note that Tc as obtained within the 3D XY model is larger than the 2D values, since fluctuations are less important in three dimensions.

Fig. 3.6. Calculated dynamical phase sti ness ns(ω)/m for a doping concentration x = 0.12 at temperatures T /t = 0.012, 0.015, 0.016, 0.017, 0.018, 0.019, 0.0195, 0.02, 0.0205, 0.021, 0.0215, 0.022, and 0.023 (from top to bottom at ω = 0). Below Tc ≈ 0.023t ≈ 67 K, Cooper pairs start to form.

the relaxation τ3 (T )−1. Note that di erent dynamics are expected when x 0.15, where T (x) → Tc and also for the overdoped cuprates, where again T > Tc , and T > Tc. Circularly polarized light might also couple to magnetic excitations in the cuprates, but then spin–orbit coupling is involved and one obtains much longer relaxation times.

Let us now turn to the important relaxation time τ2. Recently, Corson et al. have measured the complex conductivity of underdoped Bi2Sr2CaCu2O8+δ and extracted the frequency–dependent phase sti ness ns(ω)/m from their data [36]. They have found that ns(ω)/m becomes nearly independent of the frequency at a temperature given by the BKT theory. In a simple approximation this frequency should be proportional to 1/τ2 which corresponds to the typical timescale of Cooper pair phase fluctuations. Corson et al. interpret their data in terms of dynamical vortex pair fluctuations [37, 38] and conclude that vortices, and thus local Cooper pairs, should be present up to T = 100 K.

In order to investigate the timescale τ2 in more detail, we show in Fig. 3.6 our results for the dynamical phase sti ness ns(ω)/m for a doping x = 0.12 (underdoped) at various temperatures [39]. As derived in the previous chapter, the dynamical phase sti ness is related to the dynamical conductivity

where e is the elementary charge, Im σ(ω) has been obtained from the current–current correlation function and the Kubo formula using the FLEX

Fig. 3.7. Dynamical phase sti ness ns(ω)/m for the same parameters as used in Fig. 3.6, but renormalized by vortex fluctuations using (2.131) and (2.132). The dashed lines correspond to the unrenormalized sti ness. The vertical dotted line indicates the highest frequency used by Corson et al. [36].

approximation (see (2.128)) to the Green’s functions as an input. Even slightly below Tc , ns(ω)/m has a finite value, leading to the Meissner effect, followed by a redistribution of spectral weight from above twice the low-temperature maximum gap, 2∆0, to frequencies below 2∆0. This redistribution increases with decreasing T . Furthermore, we find a finite phase sti ness for ω > 0 even in the absence of Cooper pairs, i.e. for T > Tc .2

In Fig. 3.7, we show our results for the renormalized dynamical phase sti ness obtained using (2.131) and (2.132) and Dν /r02 = 1017s−1 [39]. We can clearly see that a strong renormalization of the sti ness due to Cooper pair phase fluctuations sets in at a certain frequency. The Meissner e ect is thus destroyed for all temperatures Tc < T < Tc by (slow) vortex di usion. With increasing temperature the onset of the renormalization shifts to higher frequencies. At frequencies above this onset, the vortices cannot follow the field and thus do not a ect the response. However, the onset frequencies are always smaller than 2∆0. The structure around 2∆0 is due to Cooper pair formation and is una ected by the renormalization of the sti ness. Thus, as seen in various experiments, the strong pseudogap around ω 2∆0 evolves continuously into the superconducting gap for temperatures T < Tc.

To summarize this subsection, we have solved the generalized Eliashberg equations self–consistently and extended them by including Cooper

2 As has been discussed in Sect. 2.2, for T > Tc the e ective action of the phase vanishes. Moreover, the order parameter itself vanishes, so that its phase has no physical meaning. Thus, for T > Tc , ns (ω) is related to the normal-state skin e ect.

pair phase fluctuations to calculate some basic properties of the hole–doped cuprate superconductors. In particular, we have shown results for the weak– pseudogap temperature T , where a small reduction in the spectral density at the Fermi level appears, for the strong–pseudogap temperature Tc , where local incoherent Cooper pairs start to form, and for the superconducting transition temperature Tc, where the phases become coherent. We combined our results with standard many-body theory and used this as an input to the Ginzburg–Landau energy functional ∆F {ns, ∆}, and found a phase diagram for hole-doped cuprates with two di erent regions: on the overdoped side we obtain a mean–field–like transition and Tc ∆(T = 0), whereas in the underdoped regime we find Tc ns(T = 0).

Finally, we have compared our results with the BKT theory using FLEX data as an input and obtained similar results. We have calculated the dynamical sti ness against fluctuations, ns(ω)/m, as a function of doping and temperature, taking into account renormalization by Cooper pair phase fluctuations. We have compared our results with dynamical (time–resolved) measurements and found fair agreement. We have also reproduced the observed linear temperature dependence of 1/λ3 close to Tc, where λ n−s 1/2 is the in-plane penetration depth.

3.1.2 Electron–Doped Cuprates

It is of general interest to see whether the behavior of hole-doped cuprates described above and that of electron–doped cuprates can be explained within a unified physical picture, using again the exchange of antiferromagnetic spin fluctuations as the relevant pairing mechanism. While hole-doped superconductors have been studied intensively [40], the analysis of electron-doped cuprates has remained largely unclear. As discussed in the Introduction, one expects on general physical grounds, if Cooper pairing is controlled by antiferromagnetic spin fluctuations, that pairing with d–wave symmetry should also occur for electron–doped cuprates [41].3 Previous experiments in the last decade did not clearly support this expectation and reported mainly s– wave pairing [42, 43, 44]. Maybe as a result of this, electron–doped cuprates have received much less attention than hole–doped cuprates so far. However, phase–sensitive experiments [45] and measurements of the magnetic penetration depth [46, 47] performed recently indeed indicate d–wave symmetry Cooper pairing.

In order to obtain a unified theory for both hole–doped and electron– doped cuprates, it is tempting to use the same Hubbard Hamiltonian, taking

3 If the dominant repulsive pairing contribution in high-Tc superconductors can be described mainly by their spin susceptibility, then the underlying order parameter must change its sign. From group theory we know [25] that for a nested Fermi surface described by Q = (π, π), i.e. k+Q = − k, a dx2−y2 -symmetry order parameter is the simplest possibility.

110 3 Results for High–T Cuprates: Doping Dependence

Fig. 3.8. Phase diagram T (x) for electron–doped cuprates. The AF transition curve is taken from [48]. The solid curve corresponds to our calculated Tc values obtained from φ(k, ω) = 0. The inset shows Tc(x) for the doping region 0.18 < x < 0.12 and experimental data (squares from [49], circles from [50], and triangle from [51]). The dotted curve refers to Ts ns .

the di erent dispersions of the carriers into account of course [52]. This has aleady been discussed in relation to Fig. 2.1. For optimally doped NCCO, the Fermi surface indicated by ARPES measurements [52] and the dispersion (see (2.4))

k = −2t [cos kx + cos ky − 2t cos kx cos ky + µ/2]

have been assumed. The chemical potential µ describes the band filling. We have chosen the parameters t = 138 meV and t = 0.3. As discussed in Chap. 2, in the case of NCCO the flat band around (π, 0) is approximately 300 meV below the Fermi level, whereas for hole-doped superconductors the flat band lies very close to the Fermi level. Thus, using the resulting k in a theory of spin–fluctuation–induced pairing in the framework of the generalized Eliashberg equations using the FLEX approximation, we expect a smaller Tc for electron–doped cuprates than for hole–doped ones. Note that in the case of electron doping the electrons occupy copper d–like states of the upper Hubbard band, while the holes are related to oxygen–like p–states, yielding di erent energy dispersions as used in our calculations. Assuming similar itinerancy of the electrons and holes, the mapping onto the e ective one-band Hubbard model (see (2.1)) seems to be justified.

In Fig. 3.8, we present our results for the phase diagram Tc(x). We find, in comparison with hole–doped superconductors, smaller Tc values and superconductivity occurring in a narrower doping range, as also observed in experiments [53]. The poorer nesting properties of the Fermi surface and the flat band around (π, 0), which lies well below the Fermi level, are respon-

111

Fig. 3.9. Momentum dependence of the real part of the spin susceptibility along the path through the Brillouin zone (0, 0) → (π, 0) → (π, π) → (0, 0) at T = 100 K for ω = 0 (solid curve) and ω = ωsf ≈ 0.47t (dashed curve). The main contributions to the corresponding pairing interaction come from qpair (along the antinodes) and Qpair (along the “hot spots”) as illustrated in Fig. 3.10b.

sible for this. It turns out that the corresponding van Hove singularity lies approximately 300 meV below the Fermi level, yielding smaller Tc values than for hole–doped cuprates. Below Tc we find a dx2−y2 –wave order parameter, which will be discussed later. The narrow doping range for Tc is due to antiferromagnetism up to x = 0.13 and rapidly decreasing nesting properties for increasing x. In the inset we show a blowup of the doping region 0.19 < x < 0.12 and some several experimental data are also displayed. One can clearly see that the overall agreement between our calculated Tc(x) curve and experiment is quite remarkable. However, in the strongly underdoped regime the experiments contradict each other. Thus it is not clear whether Tc(x) should decrease. If this were be the case, one would expect Uemura scaling, i.e. Tc ns, as for hole-doped cuprates (see dotted curve).

In order to understand the behavior of Tc(x) in underdoped electron– doped cuprates, we have calculated the Cooper pair coherence length ξ0, i.e. the size of a Cooper pair, and find similar and also larger values for electron-

˚ ˚

doped than for hole-doped superconductors (from 6 A to 9 A). If owing to strong–coupling lifetime e ects, the superfluid density ns becomes small, the distance d between Cooper pairs increases. If for 0.15 > x > 0.13 the Cooper pairs do not overlap significantly, i.e. d/ξ0 > 1, then Cooper pair phase fluctuations become important [3, 4, 35]. Thus we expect, as for hole–doped superconductors, that Tc ns. Assuming that ns increases approximately linearly from x 0.13 to x 0.15, we estimate a Tc which is smaller than thr value calculated from φ(k, ω) = 0, see the dashed curve in Fig. 3.8. Thus more experiments determining Tc for x ≤ 0.15 should be performed to check the Uemura scaling Tc ns.

112 3 Results for High–Tc Cuprates: Doping Dependence d–Wave Order Parameter

In order to investigate first the underlying pairing interaction, we show in Fig. 3.9 results for the real part of the spin susceptibility at 100 K with U = 4t in the weak-coupling limit for ω = 0 (solid curve) and for ω = ωsf ≈ 0.47t (dashed curve). As discussed earlier, ωsf denotes the spin fluctuation (paramagnon) energy, where a peak in Im χ(Q, ω) occurs. The commensurate peak of Re χ(q, ω = 0) at Q=(π, π) is in accordance with recent calculations in [54], where it was pointed out that the exchange of spin fluctuations yields a good description of the normal–state Hall coe cient RH for both hole– and electron–doped cuprates. Furthermore, we also find a linear temperature dependence of the in–plane resistivity ρab(T ), if we do not take into account any additional electron–phonon coupling. This will be discussed later. Concerning the superconducting properties, note that the lower tiny peak would favor dxy pairing symmetry, but the dominant larger peak leads to dx2−y2 symmetry and is also pair–breaking for dxy symmetry. Evidently, the electron–doped cuprates are not close to dxy pairing symmetry as stated previously [55]. This explains why the resultant superconducting order parameter φ(k, ω) exhibits almost pure dx2−y2 symmetry.

In Fig. 3.10, we present our results for the superconducting order parameter φ(k, ω) calculated from the generalized Eliashberg equations for electrondoped cuprates using the FLEX approximation. We show φ(k, ω = 0) for an electron doping x = 0.15 at T /Tc = 0.8, where the gap has just opened. The gap function clearly has dx2−y2 –wave symmetry. This is in agreement with the reported linear and quadratic temperature dependences of the in–plane magnetic penetration depth at low temperatures in the clean and dirty limits, respectively [46, 47], and with phase–sensitive measurements [45]. From our result of a pure dx2−y2 -wave superconducting order parameter, we expect a zero–bias conductance peak (ZBCP) [56] as recently observed in optimally doped NCCO [57] and also in hole–doped superconductors [43]. Note that its absence in some other experiments may be attributed to small changes in the surface quality and roughness [58] or to disorder [59]. The incommensurate structure in the order parameter close to (π, 0) results from the double–peak structure in Re χ at ω ≈ ωsf = 0.47t shown in Fig. 3.9. This means physically that the Cooper pairing interaction occurs mostly not for a spin–fluctuation wave vector Q = (π, π), but mostly for ω = ωsf and Q = (π − δ, π + δ). Furthermore, from Figs. 3.9 and 3.10b we conclude that no dxy -symmetry component is present in the superconducting order parameter, since the dominant dx2−y2 –type pairing suppresses dxy pairing. ARPES studies might test this.

On general grounds we expect a weakening of the dx2−y2 pairing symmetry if we include the electron–phonon interaction and if this interaction plays a significant role. The absence of an isotope e ect (α0 = d ln Tc/d ln M ≈ 0.05) for a doping x = 0.15 (see [60]) suggests the presence of a pure dx2−y2 symmetry. As discussed in Sect. 1.4.3, we know from Fig. 3.9 that phonons con-

Fig. 3.10. (a) Calculated dx2−y2 –wave symmetry of the superconducting order parameter at T /Tc = 0.8 for x = 0.15 in the first square of the BZ. (b) Calculated Fermi surface for (optimally doped) NCCO. The plus and minus signs and the dashed lines refer to the sign of the calculated momentum dependence of the dx2−y2 gap function φ(k, ω = 0) and its nodes, respectively.

necting parts of the the Fermi surface with wave vector Qpair = (π, π) will add destructively to the spin fluctuation pairing [61]. If, owing to exchange of spin fluctuations, a dx2−y2 -symmetry instability is the dominant contribution to the pairing interaction, an additional electron–phonon coupling with wave vector qpair = (0.5π, 0) will be pair–building. We generally expect that, owing to the poorer nesting, the pairing instabilities due to electron–phonon and spin fluctuation interactions will become more comparable. In this case

114 3 Results for High–Tc Cuprates: Doping Dependence

the electron–phonon coupling would definitely favor s–wave symmetry of the underlying superconducting order parameter. This can be analyzed in detail by adding a term α2F (q, ω) to the pairing interaction [61].

To continue the discussion of why the symmetry of the order parameter depends more sensitively on the electron-phonon interaction for electron– doped cuprates, we show in Fig. 3.10b the calculated Fermi surface for optimally doped NCCO. Note that the topology of the Fermi surface for the electron-doped cuprates is very similar to that of optimally hole-doped Bi2Sr2CaCu2O8+δ (BI2212), as was also pointed out recently in [62]. We estimate that practically no phonons are present along the edges (−0.25π, π) → (0.25π, π) bridging BZ areas where the superconducting order parameter φ(k, ω) is always positive (this condition is denoted by +/+). Attractive electron–phonon coupling bridging +/− areas, i.e. (−0.5π, −0.5π) → (0.5π, 0.5π), is destructive for dx2−y2 –symmetry Cooper pairing. However, owing to poorer nesting conditions, pairing transitions of the type +/+ contribute somewhat and then a mixed symmetry {dx2−y2 + αs} may occur 4.

Further experimental study of the doping dependence of the oxygen isotope e ect is necessary for a better understanding of the role played by the electron–phonon interaction. For example, if, owing to structural distortion and oxygen deficiency in the CuO2 plane, the phonon spectrum F (q, ω) changes significantly, then this a ects the isotope coe cient α0 and reduces Tc. Possibly the reported large isotope e ect of α0 = 0.15 for a slightly changed oxygen content, i.e. Nd1.85Ce0.15CuO3.8, could be related to this effect [63, 64]. As an example, one might think of the oxygen out-of-plane B2u mode, which becomes active if O4 is replaced by O3.8 [65]. A further signal of a significant electron–phonon coupling might be the quadratic temperature dependence of the resistivity [66].

To summarize this subsection, our unified model for cuprate superconductivity yields for electron–doped cuprates, as for hole-doped ones, pure dx2−y2 symmetry pairing, in good agreement with recent experiments [45, 46, 47]. Our results seem physically clear in view of the discussion presented in connection with Figs. 3.9 and 3.10b in particular. Moreover, the important input to the calculation, namely the dispersion k , was taken in agreement with ARPES measurements. The canonical value used for the strength of the e ective Coulomb interaction U is in accordance with this dispersion. In contrast to hole–doped superconductors, we find for electron–doped cuprates smaller Tc values owing to a flat dispersion k around (π, 0) well below the Fermi level. Furthermore, superconductivity occurs only for a narrow doping range 0.18 > x > 0.13, because of the onset of antiferromagnetism and, on the other side, poorer nesting conditions. We obtain 2∆/kB Tc = 5.3 for x = 0.15 in reasonable agreement with experiment [42]. We argue that if the electron–

4 Owing to our tetragonal ansatz, one component of the resulting order parameter must be imaginary, e.g. {dx2−y2 +iαs}. However, a slight orthorhombic distortion would allow our proposed {dx2−y2 + αs} symmetry.

phonon coupling becomes important, for example owing to oxygen deficiency, then the s–wave pairing instability competes with dx2−y2 -wave symmetry. This might explain the possible s–wave symmetry order parameter reported in earlier measurements.

3.2 Elementary Excitations in the Normal

and Superconducting States: Magnetic Coherence, Resonance Peak, and the Kink Feature

In this section we assume the exchange of antiferromagnetic spin fluctuations to be the relevant Cooper pairing mechanism and calculate the elementary excitations of the spin and charge degrees of freedom in high–Tc superconductors. In most cases it is also possible to describe the temperature dependence and doping dependence. Moreover, we shall study the consequences of the important feedback of superconductivity on the elementary excitations mentioned in Chap. 2 and the relationship between di erent experimental techniques. Thus, in short, we shall present many fingerprints of spin–fluctuation–mediated pairing that can be seen in the experiments.

3.2.1 Interplay Between Spins and Charges:

a Consistent Picture of Inelastic Neutron Scattering Together with Tunneling and Optical–Conductivity Data

If antiferromagnetic spin fluctuations are the main pairing mechanism in high–Tc superconductors, it is important to understand the spin–excitation spectrum as observed by inelastic neutron scattering [67, 68]. This means, in particular that the doping and temperature dependences of the spin susceptibility Im χ(q, ω) and their relationship to the superconducting transition temperature Tc are important. INS experiments show the appearance of a resonance peak at ωres only below Tc [67] and find a constant ratio of ωres/Tc 5.4 for underdoped YBa2Cu3O7−δ (YBCO) and overdoped Bi2Sr2CaCu2O8+δ (BSCCO) [68, 69, 70]5. Furthermore, recent INS data on La2−xSrxCuO4 (LSCO) reveal strong momentum– and frequency– dependent changes of Im χ(q, ω) in the superconducting state [71, 72], which the authors called magnetic coherence e ect. In particular, Im χ(Qi) for Qi = (1 ± δ, 1 ± δ)π is strongly suppressed compared with its normal–state value below ω < 8 meV, while it increases above this frequency. Moreover, the incommensurate peaks become sharper in the superconducting state [71, 72].

Our aim in this section is to use an electronic theory for the spin susceptibility and for Cooper pairing via exchange of antiferromagnetic spin

5A closer inspection for the normal-state data of underdoped YBa2Cu3O6+x [69] shows that this peak is qualitatively di erent from the resonance peak [70].

116 3 Results for High–Tc Cuprates: Doping Dependence

fluctuations to analyze the consequences of the feedback of superconductivity for magnetic coherence and the resonance peak, and on the relationship between INS, tunneling, and optical conductivity. Using the RPA and self– consistent FLEX [73] calculations of the generalized Eliashberg equations for Im χ(q, ω), we present results for the kinematic gap (or spin gap) ω0, and for ωres, ωres/Tc, and the gap function ∆(ω) in reasonable agreement with experiments. Thus, most importantly, we find that our electronic theory in the framework of the generalized Eliashberg equations can explain consistently the INS, optical–conductivity, and SIN tunneling data. Moreover, the same physical picture gives results for both underdoped and overdoped cuprates [74, 75, 76]. We find that the resonance peak in the magnetic susceptibility Im χ(q, ω) appears only in the superconducting state, that it scales with Tc, and that magnetic coherence is a result of a d–wave order parameter.

BCS–Like Analysis of the Spin Susceptibility:

Resonance Peak and Magnetic Coherence

In order to analyze the kinematic gap and the position of the resonance peak, it is instructive to start with the bare BCS susceptibility [77]

where its imaginary part reads at q = Q = (π, π)

dispersion of the Cooper pairs in the superconducting state. In the following we use a gap function with d–wave symmetry, ∆k = ∆0(cos kx − cos ky )/2, which can be calculated self-consistently within our FLEX-approach. For the normal–state dispersion, we employ the tight–binding band introduced in

(2.4),

k = −2t [cos kx + cos ky − 2t cos kx cos ky − µ/2] .

Here, t is the nearest–neighbor hopping energy, t denotes the ratio of the next–nearest–neighbor to the nearest– neighbor hopping energy, and µ is the chemical potential. We use t as a fitting parameter in order to describe the

Fermi surface topology of the two materials YBCO and LSCO. In evaluating (3.4) we have taken t = 0 and, for simplicity, have not considered a bilayer coupling via a hopping integral t [78]6.

As can already be seen within a BCS-like approach, the susceptibility Im χ0(Q, ω) involves two characteristic frequencies. The first, ωDOS , arises from the density of states of the Bogoliubov quasiparticles (i.e. the Cooper pairs), which have a gap in their spectrum due to superconductivity, ωDOS 2∆(x, T ). Here and in the following, x is the doping concentration. The second frequency, ω0, at which Im χ0(Q, ω) starts to increase represents the existence of a d–wave superconducting order parameter and is the socalled kinematic gap [77, 78]. Note that using the full FLEX approach, we find that the kinematic gap is washed out for t > 0.3.

In order to discuss both the resonance peak and magnetic coherence, we show in Fig. 3.11a results for the spin susceptibility Im χ(Q, ω) defined in (2.125). We again obtain the two characteristic frequencies ω0 and ωres ωDOS at which Im χ is peaked. Furthermore, one can clearly see that with increasing U the peak in Im χ shifts to lower energies and, most importantly,

this signals the occurrence of a spin–density–wave collective mode. The real part is given (at T = 0) by

Re χ0(Q, ωres) has been investigated in detail in [79], where it was found that the spin–density–wave collective mode, which satisfies (3.5), can explain the dip and hump feature observed in the photoemission spectra of BSCCO [80]. In particular, it was shown that the broad humps are at the same position for both the normal and the superconducting state.

We find from (2.125) that, in the normal state where no resonance appears, the spin wave spectrum is mainly determined by the spin fluctuation frequency ωsf (roughly the peak position) and, for q = Q by the Ornstein–

On the other hand, in the superconducting state one finds that Im χ peaks resonantly at ωres, where ωres 2∆, as can already be seen from (3.4). More precisely we find for optimal doping, where (3.5) determines the structure of Im χ the important relation ωres(T ) ≈ 2∆0(T ) − ωsf (T ). Physically speaking, the resonance peak peak that appears in INS only below Tc is mainly

6Bilayer coupling leads to better nesting of the bonding and antibonding bands [89]. Thus a resonance peak might appear for a complicated band structure also.

118 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.11. Numerical results for the resonance peak and magnetic coherence in the weak-coupling limit. (a) Imaginary part of the RPA spin susceptibility (in units of states/eV) versus ω in the superconducting state at wave vector q = Q = (π, π) for U/t = 1, 2, 3, and 4 (from bottom to top). As in [67], we find ωres = 41 meV. Below the kinematic gap ω0, Im χ(Q, ω) is zero. (b) Calculated magnetic coherence: the solid curves correspond to the superconducting state and the dotted curve to the normal state. The four peaks observed occur at Qi = (1 ± δ, 1 ± δ)π, and in the figure we show only the peaks at Qi = (1, 1 ± δ)π. In our calculations we find δ ≈ 0.18. These results are in fair agreement with experiments, see [71, 72].

determined by the maximum of the superconducting gap, but is renormalized by normal–state spin excitations. This provides a simple explanation for the observed 41 meV resonance peak in optimally doped YBCO [67], because Raman data suggest 2∆ = 58 meV [81], and ωsf 17 meV (at 100 K) as extracted from NMR experiments [82].

We show in Fig. 3.11b results for the q dependence of Im χ(q, ω) obtained using the same BCS–like analysis. We performed our calculations for U = 2t and a superconducting gap of 2∆ = 10 meV, as measured by Raman scattering in optimally doped La1.85Sr0.15CuO4 [83]. For ω = 10 meV, we obtain two peaks at q = Qi. In the superconducting state, we find a sharpening of

the peaks due to the occurrence of a gap. This simply means that the lifetime of the quasiparticles is enhanced owing to a reduced scattering rate. At 4 meV these peaks are strongly suppressed, as seen in experiments [71, 72]. Moreover, we find no signal for ω < 4 meV. This is due to the kinematic gap seen in Fig. 3.11a which is independent of q. Note that the situation were to be totally di erent if LSCO were to have an isotropic gap where all states for 0 < ω < 2∆0 20 meV were forbidden. In this case no kinematic gap (or spin gap) would be observed.

We conclude from the above analysis and from Fig. 3.11 that, even in the weak–coupling limit where no lifetime of the Cooper pairs (i.e. ∆ is independent of ω) is considered, we are able to explain the resonance peak and the magnetic coherence e ect within a unified picture using a dx2−y2 – wave order parameter. However, on this level no microscopic justification for a d–wave order parameter can be given. In particular, its ω dependence will also be important.

Feedback E ect of Superconductivity on the Spin Susceptibility: Resonance Peak

In order to consider the important feedback e ect of ∆ on the spin excitation spectrum, we now discuss our results obtained in the strong-coupling limit (i.e. ∆ is ω–dependent) by solving self–consistently the generalized Eliashberg equations within the FLEX approximation [73, 84]. Note that only U/t and the tight–binding dispersion relation (k) (with its band filling µ) enter the theory as free parameters. We further assume a rigid–band approximation.

In Fig. 3.12a we present results for Im χ(Q, ω) calculated for U = 4t and an optimum doping concentration x = 0.15 which corresponds to µ = 1.65 in (2.4). In the normal state (short–dashed curve) we find roughly the spin fluctuation energy ωsf = 0.1t, whereas for T < Tc the resonance peak (solid curve) appears at ωres = 0.15t. The long–dashed curve corresponds to T = 0.9Tc, where the superconducting gap starts to open. Thus, the peak position reveals information about the temperature dependence of the superconducting gap. For temperatures T < 0.75Tc the resonance peak remains at ωres = 0.15t and only the peak height increases further. We find that the height of the peak is of the order of the quasiparticle lifetime 1/Γ (ωres), where Γ (k, ω) = ω Im Z(k, ω)/Re Z(k, ω); Z denotes the mass renormalization within the Eliashberg theory. Thus we can conclude that the resonance peak becomes observable because the scattering rate decreases drastically below Tc [79].

In order to relate ωres to ∆, we show in Fig. 3.12b the corresponding calculated density of states N (ω). Below T < 0.75Tc we find that the value of 2∆ determined from the peak–to–peak distance stays approximately constant and is very close to the value ωres seen in INS, i.e. 41 meV, as shown in Fig. 3.12a. This is in good agreement with measured STM SIN tunneling data in [85]. However, in SIN tunneling a renormalized value of 2∆ is observed.

120 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.12. Consistent picture of INS and tunneling data. (a) Imaginary part of the RPA spin susceptibility at q = Q = (π, π) calculated within the FLEX approximation for optimum doping x = 0.15. For the normal state (short–dotted line) we obtain ωsf = 0.1t and for the superconducting state we obtain ωres = 0.15t. Assuming t = 250 meV we find that 0.16t = 40 meV. Inset: imaginary part of the gap function at T = 0.7Tc for wave vector q (π, 0). (b) Calculated density of states for the same parameters and temperatures as in (a).

Note that a direct measurement of ∆(ω) (e.g. by SIS tunneling) would lead to higher values. For example, we show in the inset of Fig. 3.12a the imaginary part of the gap function at the wave vector q (π, 0), where the gap has its maximum. It is peaked at ω = 0.25t.

On the other hand, we show in Fig. 3.13 our results for the feedback of superconductivity for the (optimally doped) electron–doped superconductor Nd2−xCexCuO4 (NCCO), obtained using the tight–binding energy dispersion shown in Fig. 2.1 as an input. Clearly, a rearrangement of the spectral weight occurs for small frequencies, but no resonance peak is present. The rearrangement is again due to the structure in ∆(ω), which, however, occurs at smaller frequencies because the gap is smaller than in hole-doped superconductors. We obtain no resonance peak for NCCO because the resonance condition (3.5) cannot be fulfilled. Thus we conclude that the occurrence

Fig. 3.13. Calculated imaginary part of the spin susceptibility for the electron– doped superconductor NCCO above and below Tc at q = Q = (π, π) obtained using the FLEX approach. The calculations were performed for U = 4t and an optimum doping concentration x = 0.15. The band structure was taken from experiment (see Fig. 2.1).

of a resonance peak is not proof of spin–fluctuation–induced pairing in the cuprates; the absence of a resonance peak can also be explained within our theory! Instead, the resonance peak should be viewed as an important fingerprint of spin fluctuations. On the other hand we find that a rearrangement of the spectral weight is always present, reflecting the structure in ∆(ω) and thus the character of the spin excitations themselves itself (for example, their energy ωsf ). Note that in our electronic theory, there is no need for a second CuO2 plane in order to describe the resonance peak. In our view, a second plane provides another possible way to satisfy (3.5), but is not needed in general. Thus we also expect a resonance peak in the single–layer thallium compound. This indeed has been observed recently [86].

Let us come back to the hole–doped cuprates, where a resonance peak is clearly present. In order to discuss the consequences of our analysis for the optical conductivity and, in particular, the consequences of the feedback of superconductivity on Im χ for various superconducting properties we have derived the following result (see Appendix B)7

k

7We have calculated the self-energy of an electron due to spin fluctuations in the lowest order. We have assumed further that the main contribution to the momentum sum comes from the nesting vector Q.

122 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.14. Calculated scattering rate in the normal state for T = 1.5Tc (dashed curve) obtained using (3.8) and in the superconducting state at 0.75Tc (solid curve) obtained using the Kubo formula [20, 87], yielding a threshold (2∆ + ωres). The results are in fair agreement with [88].

where N (ω) = k δ(|ω|− Ek) is the density of states. Equation (3.8) is valid in both the normal and the superconducting state. It permits discussion of how much Σ(ω) reflects ωsf and ωres, for example. We see that the feedback of superconductivity on Im χ causes, approximately, a shift of the elementary

U Im χ(Q, ω)/4. Using (3.7) in (3.8) would not take into account the important feedback of superconductivity on Im χ. Equation (3.8) can be used to demonstrate the relationship between INS and optical conductivity measurements for the normal state, but note that for the superconducting state we calculate σ from (GG + F F ) [20]. Using Drude theory, we find that the scattering rate τ −1(ω) (which is a two–particle quantity) agrees qualitatively with −2 Im Σ(ω) for the normal state. However, in order to obtain quantitative agreement with experimental data one has to use τ −1(ω) = Γ (Q, ω) or the Drude formula. This is shown in Fig. 3.14, and the results are in good agreement with [88]. From this analysis we can conclude that the optical– conductivity data, the lifetime of the quasiparticles, the resonance peak, and the SIN tunneling data can be understood within our electronic theory.

Doping Dependence of the Resonance Peak

As mentioned above, the doping dependence of the resonance peak is of significant interest for understanding the spin excitations in high–Tc cuprates. In Fig. 3.15, we show results for ωres as a function of the doping concentration. We find that, for a fixed U , (3.5) cannot be fulfilled in the overdoped case8. Thus we find that in this regime, the resonance peak is determined

8Note that away from optimal doping, by determining formally the minimum of 1 − U Re χ0 one obtains ωmin = 2∆0 − ωsf . However, the physically relevant condition 1 = U Re χ0 yields ωres.

Fig. 3.15. Calculated results for the resonance frequency ωres versus doping, obtained using the FLEX approximation. In the overdoped regime, where Tc ∆0 [76], we find a constant ratio ωres/Tc 8.

mainly by Im χ0(Q, ω) and thus by 2∆0. On general grounds one expects Tc ∆0 in the overdoped regime, where the system behaves in accordance with mean–field (BCS) theory. This has been discussed in the previous section and in [76]. Thus we conclude that ωres/Tc should be a constant ratio. We find ωres/Tc 8, which is larger than the observed value in BSCCO [68]. This is due to an underestimation of Tc within the FLEX approximation and to phonons, which are neglected in our work.

In contrast to the overdoped case, we find in the underdoped regime, where Tc ns (ns denotes the superfluid density) [76], that the resonance condition (3.5) yields ωres ωsf , which decreases. Note that the superconducting gap guarantees that (3.5) is fulfilled. Thus we find a decreasing resonance frequency for decreasing doping, in agreement with earlier calculations

where the optimally doped case corresponds to xopt = 0.15 holes per copper site. This predicted doping dependence of the resonance peak position should be tested further experimentally.

In short, we are able to explain consistently all characteristic facts about the spin excitation spectrum of high–Tc cuprates seen in INS and its doping dependence within an electronic theory using the generalized Eliashberg equations. In particular, we find that the resonance peak is a rearrangement of the spectral weight of the normal state which happens only below Tc. Thus it is rather di cult to reconcile the resonance peak with the stripe picture, for example. Furthermore, we have shown that magnetic coherence is connected with the resonance peak and can be explained by a kinematic gap

124 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.16. Self–consistent scheme used by Carbotte et al. in order to describe the coupling of quasiparticles to spin fluctuations using conductivity scattering rates and Eliashberg equations within the semiclassical approximation. This procedure has been successfully applied for many hole–doped cuprates [75].

and d–wave symmetry of the superconducting order parameter. By taking into account the feedback of superconductivity on Im χ(q, ω) we argue that the ARPES results, the tunneling data, and the measurements of the optical conductivity are consistent.

Comparison with Other Approaches

Carbotte and Schachinger made also look an important step towards a unified description of the optical conductivity of hole–doped cuprates and their spin spectrum seen in INS [75, 91]. They extended some work by Marsiglio et al. [92], in which the pairing potential W (ω) is related to the optical scattering rate τ −1(ω) by

The procedure used is shown in Fig. 3.16. Using the extracted pairing potential (shifted by ∆0) as an input to the Eliashberg equations within the semi–classical approximation (i.e. restricted to the Fermi surface) Carbotte et al. calculate Tc and a new scattering rate τ −1(ω). This leads to a new pairing potential, which has to be compared with the original potential extracted from experimental data. After changing parameters in the Eliashberg